Algorithms behind the Correlation Setting Window

Transcription

1 Algorths behd the Correlato Settg Wdow Itroducto I ths report detaled forato about the correlato settg pop up wdow s gve. See Fgure. Ths wdow s obtaed b clckg o the rado butto labelled Kow dep the a scree of Statool. We lst the approaches whch are used to calculate the theoretcal correlato ad epectato of. Also we deostrate how to get costrats for lear prograg fro the dfferet settg values. Obtag Epectato of Based o the Jo Dstrbuto: E t Ths secto llustrates how to fgure out the possble rage of the epectato of f the argal dstrbutos of ad are kow. Assue that the argal dstrbutos of ad are kow as lsted the followg table. ] ] [ l h [ l h [ l h ] [ l h ] Accordg to the defto of E we have E terval values. Based o terval ultplcato [ l h ][ l h ] [ a. Here ad are l l l h h l h h l l l h h l h h ] Let M l l l h h l h h Ma a l l l h h l h h The E [ M Ma ] ad There also are the costrats o the s fro the argal dstrbutos. These are the rows ad colus costrats as follows; for to

2 for to Here ol the to to are ukow. Our obectve s to fd the u ad au values possble for E. Sce each s o-egatve the u value of E s obtaed b zg M ad the au value of E s obtaed b azgma. Therefore two lear progras are costructed to get the u ad au values of E. Mu value: Mze M Subect to: for to for to Mau value: Maze Ma Subect to: for to for to After solvg these two lear progrags the u ad au values of E are obtaed ad are recorded as E ad Ea. These values are preseted the Epectato of sub-wdow of the Correlato Settg pop up wdow. 3 Theoretcal Correlato Although the argal dstrbutos do t detere the eact correlato betwee two rado varables the ofte costra t to soe etet. I the followg we wll show how to copute the possble correlato rage fro the argal dstrbutos.

3 Fro the defto of correlato E E E where ad are the varaces of ad. Rearragg E E E. Fro the prevous secto the theoretcal rage of E fro the defto of E s fro E to Ea. Here we have aother forula of E fro the defto of correlato. We cosder coputg the possble rage of E fro ths ew defto. E ca be wrtte as E E E E E We defe the fucto F Ths s a terval-valued fucto. We wrte the correspodg real fucto as F where to to ad ] [. I ths fucto there are varables ad ever varable s restrcted to the specfed terval rage. We ca use a optzato ethod to fd the u ad au value for F ad record the as F ad Fa. Ths s a olear optzato proble. Now we get two rages for E fro the dfferet forulas. Sce both are true we eplot both b tersectg the. Call the low ad hgh bouds the tersecto G ad G a. The G ad a G. The values of ad used to copute F ad F a are used aga here to copute the bouds o.

4 Sce we ust wat to get a safe rage for correlato ot ecessarl the arrowest possble rage we are doe. A ore accurate rage for correlato ca be gotte drectl fro coputg the ad E E E a of. Ths s a cople olear optal proble. Ths rage s preseted the Correlato Coeffcet Subwdow of the Correlato Settg popup wdow. 4 Mea ad ace Theoretcal rages of ea ad varace of operad are calculated b the progra. These values are drectl obtaed accordg to the deftos. Fro the defto epectato of rado varable s E. Sce s a terval value E [ l h ] l h. So the bouds o E are obtaed. The slar ethod s used to hadle operad. the bouds o E are l ad h. aces of ad are a lttle ore cople to obta. Based o the defto E E varace of s. Here each s a terval value. Ths s a proble of evaluato of a terval fucto. We defe a real fucto V ad each to. Sce all are kow the optzato ethod ca be adapted to copute the ad a values of fucto V as V ad V a. The slar ethod s used to varace of. Let V ad to. The the bouds of varace of are obtaed recorded as V ad V a. These rages are preseted the Mea ad ace Subwdow of the Correlato Settg popup wdow. 5 Costrats fro settg the rage of correlato I ths secto we deostrate how to get etra costrats f the user sets the rage of correlato the Correlato Coeffcet Subwdow of the Correlato Settg popup wdow.

5 Fro secto E [ M Ma ] M Ma sce s o-egatve. Fro secto 3 E F Usg the real fucto F ad to to ad gve rage for correlato the u ad au values of F ca be calculated b o-lear optzato as secto 3. Call the F ad Fa. Based o Berleat & Zhag [] two equaltes are defed: M F a ad Ma F. These two equaltes for two etra costrats for lear prograg sce ol the s are ukow. 6 Costrats fro settg the rage of E If the user sets E rage the Epecto of E subwdow of the Correlato Settg popup wdows the values that the user proves F ad Fa are used drectl to defe costrats: M Ma F a F. These costrats were ustfed the secto 5 ad Berleat & Zhag []. 7 Costrats fro settg Mea ad ace of ad The user ca set ea ad /or varace the Mea ad ace Subwdow of the Correlato Settg popup wdow. Cosder the forula E E E. If the eas ad varaces of ad are kow the value of E ca be calculated f correlato s also kow. Fro secto 5 the rage for correlato s coputable. We ca use ths rage of correlato to calculate the rage of E. It s clear that coputg E s terval ot a real uber. Let the low boud of E be called F ad the hgh boud be called Fa. The

6 M used b Statool. F a ad Ma F. These costrats are the 8 Costrats fro Settg Correlato Mea ad ace of ad I the soe stuatos the user a kow partal forato about both correlato ad ether ea varace or both. Here s how the user ca choose values for correlato ad ea ad/or varace. Frst the user should clck o the checkbo butto labelled Iput data both the correlato subwdow ad the ea ad varace subwdow the Correlato Mea ad ace Subwdow of the Correlato Settg popup wdow. The the user ca set values both the Correlato ad Mea ace subwdows of the Correlato Settg popup wdow. I secto 7 we descrbe the stuato where ea ad/or varace are kow. If correlato s also put we ca drectl use all three the forula E E E to get the value of E. If ether ea or varace s ssg a default rage for t a be obtaed as descrbed secto 4. Let the low boud of E be called F ad the hgh boud be called Fa. The M are the added to the L calls. F a ad Ma F. These etra costrats 9 Refereces [] D. Berleat ad J. Zhag Usg correlato to prove evelopes aroud derved dstrbutos Relable Coputg press as of 3/7/03

7 Fgure. "Correlato Settg" popup wdow.